Speeding up the Euler scheme for killed diffusions

Abstract

Let $X$ be a linear diffusion taking values in $(\ell ,r)$ and consider the standard Euler scheme to compute an approximation to $\mathbb{E}[g(X_T)1_{[T<\zeta ]}]$ for a given function $g$ and a deterministic $T$, where $\zeta =\inf \{t\ge 0:X_t\notin (\ell ,r)\}$. It is well-known since \cite{GobetKilled} that the presence of killing introduces a loss of accuracy and reduces the weak convergence rate to $1/\sqrt{N}$ with $N$ being the number of discretisatons. We introduce a drift-implicit Euler method to bring the convergence rate back to $1/N$, i.e. the optimal rate in the absence of killing, using the theory of recurrent transformations developed in \cite{rectr}. Although the current setup assumes a one-dimensional setting, multidimensional extension is within reach as soon as a systematic treatment of recurrent transformations is available in higher dimensions.

Some typos and errors in the earlier version are corrected